A new trimester will begin in just a couple of weeks, and I will have a new bunch of geometry students. Before long, we'll be digging in to geometric proof, which is a concept/activity/process that we will develop and use throughout most of the course. It is always a tough concept to develop with students, and I am always wondering how I can make it easier for them to grasp.
One really nice, fairly accessible activity that we have used with students at my school is "The Letter Game", which was published by Don Gernes in his article The Rules of the Game (full citation below). The game amounts to a very simple deductive system, in which students are asked to "prove" various statements. It gives students an opportunity to practice working within the bounds a deductive system, and to get a sense for the structure of proofs and the process of proving something. This comes before they have to confront the added complexity of geometry, which is the big deductive system they ultimately will be learning to work within.
Before getting to the "The Letter Game", Gernes suggests having students think about games that they are already familiar with: monopoly, basektball, soccer, etc. Each of these games has undefined terms, defined terms, postulates, and maybe even some theorems. These are the makings of a deductive system. Gernes presents basketball as an example, and here's what he lists in each of these categories:
______________
Undefined Terms
Ball
Player
Court
Baskets
Defined Terms
Field Goal
Foul
Free Throw
Traveling
Postulates
If a player is fouled, then the player gets to shoot a free throw.
If a player travels, then the other team gets possession of the ball.
If a player makes a field goal, then the player's team gets two points.
Theorem
The referee objectively applies the rules of the game to each play.
This obviously doesn't represent a complete list of all of the terms and rules that make up the game of basketball, but it is enough to have a discussion about what the undefined terms, etc. are, and how each of them contributes to the structure of a deductive system.
The Letter Game is where students get a chance to try their hand at some proofs. Gernes keeps the system simple, so students don't have too much to keep track of as they begin proving "theorems". Here's the deductive system he establishes:
______________
Undefined Terms
Letter M, I, and U
Definition
x means any string of I's and U's.
Postulates
1. If a string of letters ends in I, you may add a U at the end.
2. If you have Mx, then you may add x to get Mxx
3. If 3 I's occur, that is, III, then you may substitute U in their place.
4. If UU occurs, you drop it.
With these four postulates, students take given strings of letters and then "prove" another string of letters, using one postulate at a time to manipulate the string. For example - Given: MIII Prove: M.
Even though many of my students end up groaning about geometric proofs, most of them end up enjoying the letter game. It is accessible enough that most of them are able to make some headway, and after a quick example or two, most of the class is able to work independently. I usually don't get any students reverting to "shut down" mode because they can't get it. Sometimes I even have students finished early who create their own new letter "theorems" and challenging one another to prove them. While the activity doesn't make them masters of geometric proof, it does reinforce the concept of a deductive system, and what it is to work in and prove new theorems in a deductive system. Every step that is taken or statement that is made needs to be backed up by a definition, postulate, or theorem.
Opening up the much more broad deductive system of geometry, with all of its definitions, postulates, and theorems, always makes geometric proof more difficult for my students. The Letter Game hasn't eliminated those difficulties for my geometry students, but it has been a good first experience with proof for them. Gernes' activities build a nice segue from concepts that students are familiar and comfortable with to the way geometry works as a system. His article is definitely worth a look:
Gernes, D. (1999). The rules of the game. The Mathematics Teacher, 92(5), 424-429.
Wednesday, February 26, 2014
Wednesday, February 5, 2014
An Introductory GeoGebra Activity for Calculus Students
My calculus students had never used GeoGebra before this activity, so I wanted to do something fairly simple with them. This activity takes about five steps on GeoGebra, so it wasn't too demanding for first time users. At the same time, it provided some nice visual support for the concepts we had been learning in class, and provided them with a tool to explore those concepts further. I've tried to make the instructions below friendly for first time GeoGebra users, and I'm a novice anyway.
1) First, we entered a function in the Input bar at the bottom of the window. The function we started with was y = x^3+3x^2-1. In the algebra pane, GeoGebra relabels this as f(x). This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points). At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.
2) With our function graphed, we added a new point on the curve by selecting the Point tool and clicking on the curve. GeoGebra automatically labels this point A.
3) We then put a tangent line on the curve at point A. GeoGebra has a tool for this under the fourth box from the left. Once the Tangents tool is selected, click on the curve and click on the point of tangency (point A for us), in no particular order. This is a good time to show students how to use the Move tool if they haven't discovered it already. It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool). If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.
4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative). GeoGebra has a variety of measuring tools. Once the Slope tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope. The value of the slope gets added to the algebra pane as m.
5) We finally add one more point to the sketch, using the Input bar again. Type in (x(A),m) to define this last point, which GeoGebra will call point B. This tells GeoGebra to use the x-value from point A (the point on the curve) and to plot the slope of the curve as the y-value. This point displays the value of the graph's derivative at any x-value. By right-clicking on point B, Trace On can be selected.
With trace turned on, the path of this point will be traced out on the sketch as point A is moved along the curve, revealing the shape of the derivative function. (I also changed the color of point B using the pull-downs at the top of the graphics view.)
Setting Within the Course
When I brought my class to the computer lab for this activity, my calculus students had already been using the limit definition of a derivative for three days. We had also been discussing the derivative at a point as the function's instantaneous rate of change, and the slope of a line tangent to the function at that point. When we had opportunities to do so, we were taking special notice of times when the derivative was zero and the tangent line was horizontal. We had also spent just a little bit of time more generally discussing the relationships between graphs of functions and the graphs of their derivatives, including a few mentions of concavity. Furthermore, after working with the limit definition for several days, my students had noticed and we had informally discussed the power rule.The Activity
With these concepts in mind, we did the following together on GeoGebra:1) First, we entered a function in the Input bar at the bottom of the window. The function we started with was y = x^3+3x^2-1. In the algebra pane, GeoGebra relabels this as f(x). This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points). At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.
2) With our function graphed, we added a new point on the curve by selecting the Point tool and clicking on the curve. GeoGebra automatically labels this point A.
3) We then put a tangent line on the curve at point A. GeoGebra has a tool for this under the fourth box from the left. Once the Tangents tool is selected, click on the curve and click on the point of tangency (point A for us), in no particular order. This is a good time to show students how to use the Move tool if they haven't discovered it already. It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool). If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.
4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative). GeoGebra has a variety of measuring tools. Once the Slope tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope. The value of the slope gets added to the algebra pane as m.
5) We finally add one more point to the sketch, using the Input bar again. Type in (x(A),m) to define this last point, which GeoGebra will call point B. This tells GeoGebra to use the x-value from point A (the point on the curve) and to plot the slope of the curve as the y-value. This point displays the value of the graph's derivative at any x-value. By right-clicking on point B, Trace On can be selected.
With trace turned on, the path of this point will be traced out on the sketch as point A is moved along the curve, revealing the shape of the derivative function. (I also changed the color of point B using the pull-downs at the top of the graphics view.)
Discussion
The completed sketch opens up a lot of good talking points. Students should be able to see that the local maximum and local minimum of f(x) line up with the zeros of the derivative function. If inflection points have been discussed, it's also nice to see that this point lines up with the vertex of the derivative. As point A is moved back and forth across the inflection point, the tangent line makes a nice rocking motion that shows the slope is beginning to increase/decrease. In general, the tangent line in the sketch provides a good visual connection between the curve of f(x) and the sign of the derivative.
Of course, students can also try to predict the shape of the derivative function, or find the derivative function algebraically and plot it on the sketch as well. Wherever possible, it seems good to keep students' visual and algebraic understandings closely intertwined. (Check out Aspinwall and Shaw's (2002) article, When Visualization is a Barrier to Mathematical Understanding, in Mathematics Teacher vol. 95, no. 9)
This GeoGebra sketch is also a good tool for students to explore derivatives for functions that they don't yet know how to differentiate algebraically. Helping my students work through this activity took less time than I had anticipated, so they had about ten minutes of class time in the computer lab after we were done with the first function. After I explained that they could change f(x) by double-clicking it in the algebra pane without altering any of the other features of the sketch, most of them started exploring other functions. Students were discovering on their own that cosine is the derivative of sine, and one of them was looking at the graph of tangent, and noticing similarities (but important differences) between the sketched derivative and the secant function. Having a tool available to explore derivatives increased their curiosity. I may pull up the same sketch again in class as we learn about the derivatives of specific types of functions.
If you would like to go to the completed sketch, here is the GeoGebra Tube link.
Monday, January 20, 2014
My Teaching Situation
I am currently teaching Geometry, Algebra II, and Calculus (non-AP). Aside from my Calculus class, my classes are all fairly sizable, at 26, 28, and 30 students. Calculus, on the other hand, has only 10 students in it, so I get to know those students especially well. There were more signed up at the beginning of the year, but I think many of those students made the jump to AP Calc before the term began.
We are on trimesters, with 5 class periods a day (we call them "macros"). Each class period is 72 minutes long. We have a great building and a lot of resources available. I have a document camera and projector in my room. We do not have one-to-one technology, but there are laptop carts and computer labs available when I have a GeoGebra activity to do with my students.
One aspect of my teaching situation that has been a little bit challenging at times is the distribution of our math department throughout the building. There are two other math teachers in my wing of the building, but the other eight of them are spread out in other locations. When I am teaching the same class with those other teachers, I have to be deliberate about checking in with them and collaborating with them. There would be a lot more impromptu collaboration if the departments in our building were arranged by wing. I know others in Math 629 have mentioned much more real isolation, though, so this is only a minor challenge.
Another challenge, although I know it's not the least bit unique, has always been time. I feel this particularly in Geometry. Both halves of our Geometry course fit very tightly within a trimester.
We are on trimesters, with 5 class periods a day (we call them "macros"). Each class period is 72 minutes long. We have a great building and a lot of resources available. I have a document camera and projector in my room. We do not have one-to-one technology, but there are laptop carts and computer labs available when I have a GeoGebra activity to do with my students.
One aspect of my teaching situation that has been a little bit challenging at times is the distribution of our math department throughout the building. There are two other math teachers in my wing of the building, but the other eight of them are spread out in other locations. When I am teaching the same class with those other teachers, I have to be deliberate about checking in with them and collaborating with them. There would be a lot more impromptu collaboration if the departments in our building were arranged by wing. I know others in Math 629 have mentioned much more real isolation, though, so this is only a minor challenge.
Another challenge, although I know it's not the least bit unique, has always been time. I feel this particularly in Geometry. Both halves of our Geometry course fit very tightly within a trimester.
Monday, December 9, 2013
Euclid's Elements - Book III
At this point, I am finished with my project - a translation of Book III of the elements with links to GeoGebra manipulatives (no proofs at this point, however). Please take a look, and feel free to leave feedback.
You can use the link to the right under Pages, or click here.
If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, click here.
Big Take-Aways from Math 641
As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:
Get students asking questions.
As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.
GeoGebra is an excellent tool and resource.
I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.
There are far more ways to present and explore mathematical ideas than I am currently aware of.
That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like Mathematics Teacher, resources like the mathtwitterblogosphere are another great place to find new ideas.
Geometry and mathematics in general are far more broad than most students get a chance to see.
Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing this with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.
Get students asking questions.
As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.
GeoGebra is an excellent tool and resource.
I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.
There are far more ways to present and explore mathematical ideas than I am currently aware of.
That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like Mathematics Teacher, resources like the mathtwitterblogosphere are another great place to find new ideas.
Geometry and mathematics in general are far more broad than most students get a chance to see.
Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing this with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.
Learning GeoGebra - Part 2
Working on my project for Math 641 has forced me to learn a lot more about GeoGebra. Some of the manipulatives for the propositions in Book III were quite simple, but others had a lot going on. Here are some of the new things that I learned:
How to define a figure made up of two or more other objects
I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a list. In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example
{Segment[A,B], Arc[c,B,A]}
This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a list in the algebra pane, which I could select and give a color to for emphasis. Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.
More varied ways of assigning conditions to show objects
In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,
Distance[A,B]
frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example
abs(a-b)<0.1
As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.
I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.
If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.
How to define a figure made up of two or more other objects
I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a list. In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example
{Segment[A,B], Arc[c,B,A]}
This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a list in the algebra pane, which I could select and give a color to for emphasis. Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.
More varied ways of assigning conditions to show objects
In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,
Distance[A,B]
frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example
abs(a-b)<0.1
As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.
I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.
If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.
An Interesting Idea
The basic idea is this: How to motivate students to help one another improve their understanding.
In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:
- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.
- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.
- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".
At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.
The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.
The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.
I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?
In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:
- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.
- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.
- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".
At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.
The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.
The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.
I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?
Subscribe to:
Posts (Atom)